Positive solutions in an annulus for nonlinear differential equations on a measure chain (Q2724912)

Here, the authors study the existence of (at least) two positive solutions to the second-order dynamic equation on a measure chain (time scale) \(T\), i.e., NEWLINE\[NEWLINE u^{\Delta\Delta}+f(u(\sigma(t)))=0, \quad t\in[0,1]\cap T, NEWLINE\]NEWLINE satisfying the boundary conditions NEWLINE\[NEWLINE \alpha u(0)-\beta u^\Delta(0)=0, \quad \gamma u(\sigma(1))+\delta u^\Delta(\sigma(1))=0. NEWLINE\]NEWLINE The theory of dynamic equations on measure chains unifies and extends the differential (\(T=\mathbb{R}\)) and difference (\(T=\mathbb{Z}\)) equation theories. The nonlinearity \(f\) is a positive continuous function on the positive half-line, and it is supposed to be superlinear at one endpoint (zero or infinity) and sublinear at the other (infinity or zero, respectively). This work extends the results by \textit{L. Erbe} and \textit{A. Peterson} [Math. Comput. Modelling 32, No.~5-6, 571-585 (2000; Zbl 0963.34020)], where \(f(x)\) was assumed to be superlinear/sublinear at both endpoints \(x=0\) and \(x=\infty\). The proofs of the two main results (Theorems~3.1,~3.2) are based on the corresponding Green function from the above mentioned paper by Erbe and Peterson and on an application of the Krasnoselskii fixed-point theorem in an appropriate cone of bounded functions \(x\) on the measure chain interval \([0,\sigma^2(1)]\). Similar results were independently obtained by \textit{R. P. Agarwal} and \textit{D. O'Regan} [Nonlinear Anal., Theory Methods Appl. 44A, No.~4, 527-535 (2001; Zbl 0995.34016)].NEWLINENEWLINENEWLINEThe reviewer points out certain limitations of this study, when specialized to difference equations (\(T=\mathbb{Z}\)). This is due to the choice of the measure chain interval to be \([0,1]\), which consists of only two points \(\{0,1\}\) in the discrete case. The paper will be useful for researchers interested in positive solutions to differential or difference equations, and/or in dynamic equations on measure chains (time scales).